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RICE UNIVERSITY STUDIES
Proof of Theorem 1.3. If M2 is an oriented compact two-manifold of
positive genus, then the non degenerate embeddings are an open dense
set in the set of embeddings of M2 into C2. If M2 is embedded by an ele-
ment in this dense set, by using Theorem 4.1 we find that there may be
no exceptional points (if the genus of M2 is one), or there are at least two
hyperbolic points (if the genus of M2 is greater than one). Thus the algebraic
topology does not allow us to conclude the existence of elliptic points in
this case. In [73] it was shown that there is at least one non degenerate
embedding of an oriented two-manifold of positive genus into C2, such
that at least two of the exceptional points are elliptic. Hence, we can only
conclude that there is an open (but not necessarily dense) set of embeddings
of a two-manifold of positive genus so that each such embedded manifold
has the property that its envelope of holomorphy contains a three-manifold.
Q.E.D.
5. Remarks
1. Let Mk be a real ^-dimensional differentiable manifold embedded in Ck,
к > 2. Using Thom Transversality Theory we find that there exists a dense set
of embeddings of Mk into Ck such that there are no exceptional points or the
exceptional points are a submanifold of dimension к — 2. If Mk is compact
and orientable, it was shown in [23] that Mk is totally real only if ∙∕fMk) = 0.
Otherwise, if Mk is compact and oriented and χ(Mk) ≠ 0 there exists an open
dense set of embeddings of Mk into Ck such that the exceptional points
form a submanifold of Mk of dimension k—2. Since к ≥ 3, these ex-
ceptional points are not isolated, and we cannot use the local lemma as
in the two-dimensional case. Also, if we could find that the non degenerate
embeddings are an open dense set in the set of embeddings, we have no
theorem analogous to that of Chern and Spanier to complete the process.
2. We have given an example of a two-sphere in C2 with two elliptic
points and no hyperbolic points. Does there exist a compact two-manifold
which can be embedded in C2 in such a Waythatallexceptionalpoints are
of the hyperbolic type?
3. Consider a real ^-dimensional differentiable manifold embedded in
C" where к ≥ n + 1, n > 1. A point p in Mk will be called exceptional
if dimcH l,(Mk) = к — n + 1. Elliptic and hyperbolic points can be defined
for this case and a local extension theorem similar to Theorem 2.1 of this
paper has been proved (see [7]).
4. It was shown in [14} that there is a dense open set of embeddings